reference request - Vector-valued valuations on lattices

There's a fair amount of work on valuations on (modular) lattices, by which I mean functions $v : mathcal{L} rightarrow R$ that satisfy the modular expression
$$v(x) + v(y) = v(x wedge y) + v(x vee y) $$

I'm wondering if there's been any work on vector-valued valuations (where the range of v is $R^k$ and the same relation holds) ?

In addition, I'm also interested in lower valuations (I'm not sure if this name is standard) that satisfy the submodular inequality
$$v(x) + v(y) ge v(x wedge y) + v(x vee y) $$
and possibly the generalization to $R^k$ where we replace the above by
$$v(x) + v(y) succeq v(x wedge y) + v(x vee y) $$
($succeq$ being the coordinate-wise partial order)

This is a reference request, for the most part.